Forbidden induced subgraph characterization of circle graphs within split graphs

Bonomo, Flavia; Durán, Guillermo; Pardal, Nina; Safe, Martín D.

doi:10.1016/j.dam.2020.12.021

Cited by 3 publications

(9 citation statements)

References 12 publications

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“…The third property refers to the ordering given by containment for the vertices. In [2,14], we see that every LR-row represents a vertex that is adjacent to almost every vertex in the complete partition K of G. Hence, when dividing the LR-rows into blocks, we need to ensure that each of its block is not properly contained in the neighbourhoods of vertices that are nonadjacent to at least one partition of K. Something similar holds for L-rows (resp. R-rows) and U-rows, and L-rows (resp.…”

Section: Theorem 33 ([15]mentioning

confidence: 99%

“…a partition of its vertex set into a clique K and a stable set S; we denote it by G = (K, S). In [2,14], we addressed the problem of characterizing those split graphs that are also circle by strongly relying on the characterization of 2-nested graphs proved in the current paper. We now briefly introduce the connection between split graphs that are circle graphs and 2-nested matrices.…”

Section: Introductionmentioning

confidence: 99%

“…This characterization immediately leads to a forbidden induced subgraph characterization for permutation graphs. By relying on the corresponding list of forbidden subgraphs and the fact that G is also a split graph, we concluded that G contains an induced subgraph H isomorphic to one of the graphs in Figure 1 (see [2]). Let us now consider an arbitrary split graph G (either circle or not).…”

Section: Introductionmentioning

confidence: 99%

“…We consider the (0, 1)-matrices A(S i , K i ) that represent the adjacencies between S i and K i . In [2], we show that it suffices to study each of these matrices and the relationship between them to decide whether G is a circle graph or not. More precisely, the key condition for G to be a circle graph is that each of these matrices A(S i , K i ) is 2-nested.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the key condition for G to be a circle graph is that each of these matrices A(S i , K i ) is 2-nested. This analysis allowed us to give a minimal forbidden induced subgraph characterization of those split graphs that are circle graphs [2].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

2-nested matrices: towards understanding the structure of circle graphs

Durán,

Pardal,

Safe

2021

Preprint

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A (0, 1)-matrix has the consecutive-ones property (C1P) if its columns can be permuted to make the 1's in each row appear consecutively. This property was characterised in terms of forbidden submatrices by Tucker in 1972. Several graph classes were characterised by means of this property, including interval graphs and strongly chordal digraphs.In this work, we define and characterise 2-nested matrices, which are (0, 1)matrices with a variant of the C1P and for which there is also certain assignment of one of two colors to each block of consecutive 1's in each row. The characterization of 2-nested matrices in the present work is of key importance to characterise split graphs that are also circle by minimal forbidden induced subgraphs.

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Section: Theorem 33 ([15]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

2-nested matrices: towards understanding the structure of circle graphs

Durán,

Pardal,

Safe

2021

Preprint

Self Cite

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2-Nested Matrices: Towards Understanding the Structure of Circle Graphs

Durán

Pardal

Safe

2022

Graphs and Combinatorics

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Exactly Hittable Interval Graphs

Dhannya,

Narayanaswamy,

Nisha

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.

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Forbidden induced subgraph characterization of circle graphs within split graphs

Cited by 3 publications

References 12 publications

2-nested matrices: towards understanding the structure of circle graphs

2-nested matrices: towards understanding the structure of circle graphs

2-Nested Matrices: Towards Understanding the Structure of Circle Graphs

Exactly Hittable Interval Graphs

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